Regularity theory for the spatially homogeneous Boltzmann equation with cut-off

We develop the regularity theory of the spatially homogeneous Boltzmann equation with cut-off and hard potentials (for instance, hard spheres), by (i) revisiting the L-p theory to obtain constructive bounds, (ii) establishing propagation of smoothness and singularities, (iii) obtaining estimates on the decay of the singularities of the initial datum. Our proofs are based on a detailed study of the "regularity of the gain operator". An application to the long-time behavior is presented.